Determinant of a 4×4 Matrix
Online calculator to calculate the determinant of a 4x4 matrix
Calculate Determinant
Instructions
Enter the values of the 4×4 matrix whose determinant is to be calculated. Empty fields are counted as zero. Click Calculate.
Determinant - Overview
What is a Determinant?
The determinant is a scalar value (number) that can be calculated from a square matrix. It provides important information about the matrix properties.
Formula (Laplace Expansion)
For a 4×4 matrix, the determinant is calculated using Laplace expansion along the first row:
\(\det(A) = a_{11}M_{11} - a_{12}M_{12} + a_{13}M_{13} - a_{14}M_{14}\)
where \(M_{ij}\) are the 3×3 minor determinants
Properties
- det(A) = 0: Matrix is singular (not invertible)
- det(A) ≠ 0: Matrix is invertible
- det(AB) = det(A) · det(B)
- det(AT) = det(A)
- det(kA) = k4 · det(A) for 4×4 matrices
Applications
- Test if a matrix is invertible
- Calculate the inverse matrix
- Solve systems of linear equations
- Calculate volumes in geometry
- Determine eigenvalues
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Description of the Determinant
Fundamentals
The determinant is a number (scalar) that is assigned to a square matrix and can be calculated from its elements. It is a useful tool in solving systems of linear equations.
Calculation Method:
For a 4×4 matrix, the determinant is calculated using Laplace expansion (also called cofactor expansion). The calculation involves:
- Choose a row or column (typically row 1)
- For each element, calculate its 3×3 minor determinant
- Multiply each element by its minor and sign factor
- Sum all the products
Formula
Laplace Expansion (First Row):
\(\det(A) = a_{11}\begin{vmatrix}a_{22} & a_{23} & a_{24}\\a_{32} & a_{33} & a_{34}\\a_{42} & a_{43} & a_{44}\end{vmatrix} - a_{12}\begin{vmatrix}a_{21} & a_{23} & a_{24}\\a_{31} & a_{33} & a_{34}\\a_{41} & a_{43} & a_{44}\end{vmatrix} +\)
\(a_{13}\begin{vmatrix}a_{21} & a_{22} & a_{24}\\a_{31} & a_{32} & a_{34}\\a_{41} & a_{42} & a_{44}\end{vmatrix} - a_{14}\begin{vmatrix}a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\\a_{41} & a_{42} & a_{43}\end{vmatrix}\)
Example Calculation
Example: 2×2 Matrix (Simple)
For a 2×2 matrix, the formula is straightforward:
\(\det\begin{pmatrix}a & b\\c & d\end{pmatrix} = ad - bc\)
\(\det\begin{pmatrix}3 & 8\\4 & 6\end{pmatrix} = 3 \cdot 6 - 8 \cdot 4 = 18 - 32 = -14\)
Example: 3×3 Matrix
For a 3×3 matrix, using the Rule of Sarrus:
\(\det\begin{pmatrix}1 & 2 & 3\\0 & 1 & 4\\5 & 6 & 0\end{pmatrix} = 1(1 \cdot 0 - 4 \cdot 6) - 2(0 \cdot 0 - 4 \cdot 5) + 3(0 \cdot 6 - 1 \cdot 5)\)
\(= 1(-24) - 2(-20) + 3(-5) = -24 + 40 - 15 = 1\)
Special Cases
- Identity Matrix: det(I) = 1
- Zero Matrix: det(0) = 0
- Diagonal Matrix: Product of diagonal elements
- Triangular Matrix: Product of diagonal elements
- Row of zeros: det(A) = 0
Practical Applications
Mathematics & Physics:
- Testing matrix invertibility
- Solving systems of equations (Cramer's rule)
- Calculating eigenvalues (characteristic polynomial)
- Computing volumes and areas
Engineering & Computer Science:
- 3D graphics transformations
- Testing linear independence
- Calculating cross products
- Numerical stability analysis
Key Properties of Determinants
- Multilinearity: Linear in each row/column
- Alternating: Sign changes when rows swap
- Normalized: det(I) = 1
- Multiplicative: det(AB) = det(A)·det(B)
- Transpose: det(AT) = det(A)
- Inverse: det(A-1) = 1/det(A)
- Scalar Multiple: det(kA) = kndet(A)
- Similar Matrices: Same determinant
Important Note
For 4×4 matrices, calculating the determinant by hand is quite complex! It requires calculating four 3×3 determinants, each of which requires three 2×2 determinants. This means 12 elementary calculations in total. For larger matrices, numerical methods or software are essential. The computational complexity grows as O(n!) for n×n matrices using cofactor expansion.
Learn More
For a more detailed explanation with step-by-step examples, visual aids, and practice problems, visit our comprehensive tutorial:
Detailed Description of the Determinant